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Comment by teucris | Hacker News Reader

“God made the integers, all else is the work of man”

I’m not sure if even that is true. Integer counts and succession seem like human constructs to me. An apple isn’t fundamentally a single unit. If I take a bite out of it, is it still one apple?

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This quote is neither true nor false. The author called it a "sentiment", but I think when Kronecker made this statement, he took it as some kind of principle to work out the foundation of modern mathematics.

Basically the mathematical notion of integer is trying to make sense of our idea of a discrete unit. Taking a bite of apple makes it non-unit, but I think you would agree that the notion of "one apple" or "two dogs" makes common sense.

Also when reading someone's quote, I would always consider its context. I think Kronecket had the axiomatic construction of real number in mind. The idea of rational number and real number can be investigated by just assuming the idea of increment by one is solid.

These mathematical concepts are usually covered in a undergraduate analysis class. For more information, see Cauchy's construction and Dadekind's construction.

Add: so the idea is that once we have integers, we can construct rationals and even real numbers instead of talking about reals as granted. Now when taking a bite out of an apple, we can say it takes, say 1/3, of an apple, while we only assume the idea of integers.

> assuming the idea of increment by one is solid.

Therein lies the rub! Succession is a human concept. All math is a magnificent city built upon that one stone.

I think this is a really good comment. Are you implying that the concept of integer is a consequence of the abstraction of an apple, an abstraction which is entirely human? I really like this point of view.

That’s exactly what I’m saying. I’m glad you like it!

I have been entertaining this idea for years. First time I hear someone else voice it.

Do you have an idea what the philosophical term for this idea is? It seems related to connectionism.

One might assume that abstraction and pattern matching in the brain are required to determine that two "things" are the same. It is perhaps a small step to go from comparing things to comparing integers.

Unfortunately, this line of reasoning quickly goes awry, because one cannot assume even the existence of things to describe brains or pattern matching.

The abstraction from counting to the natural numbers is called "decategorification", in category theory. The abstraction of sameness is called "equivalence" or "isomorphism".

> Unfortunately, this line of reasoning quickly goes awry, because one cannot assume even the existence of things to describe brains or pattern matching.

You might be interested in the univalent foundations approach, where the idea is that mathematics is fundamentally the study of equivalences between constructions, and equivalences themselves are first-class objects that are just as easy to talk about as any other mathematical object.

I don’t. I’ll do some research. The origin of this concept comes from discussions I had on identity, specifically the ability to separate the “self” from the rest of the universe, or to isolate some quantity of stuff and claim it’s an apple. Ultimately the universe has no concept of such boundaries.

[update] the study of this concept seems to be semiotics, which examines the creation and use of symbols. I don’t know what branch of semiotics examines the creation of the concepts of unity and succession.

Your conclusion doesn't follow your premise. Whether it's still an apple is a question of categorisation and a human construct (1). But once it's established it is indeed an Apple you can count the number of such Apples.

Counting is real and the number you count is "true" because it leads to real predictions. This tale of magic bucket is enlightening. (2)

(1): https://www.lesswrong.com/posts/aMHq4mA2PHSM2TMoH/the-catego...

(2): https://www.lesswrong.com/posts/X3HpE8tMXz4m4w6Rz/the-simple...

There are different ways to understand reality. For spacial understanding we have cartesian and euclidean models, the former suggests reality is digital, the latter suggests reality is analog, both give real predictions, both are true? For temporal understanding we have declarative and imperative models, the former suggests infinity exists, the latter doesn't, both give real predictions, both are true?

It seems you may have misunderstood my conclusion. All I meant to posit was exactly as you state:

> Whether it’s still an apple is a question of categorisation and a human construct

> Integer counts and succession seem like human constructs to me.

There's definitely some "elementary" mathematical concepts that seem native to our universe. For example, in outer space stuff tends to come together in ways that approximates spheres. We see lattices in crystalline solids all the time. There's naturally-occurring fractals all over the place. The existence of fractals -- snowflakes, trees, broccoli -- seems to "embed" a successor (to iterate over generations), so I'm not sure if it's a human construct.

There are no naturally occurring fractals, because all natural things have a minimal feature size after which quantum mechanics makes them fuzzy.

An apple is a single unit though.

An apple is made up of constituent parts, but it is a singular thing.

If you argue against an Apple being singular then you are essentially professing that nothing is singular.

When you take a bite out of an apple you now have a subset of an apple “an apple with a bite out of it”.

To me saying God made the integers is more akin to saying, there is and there isn’t, and all else is built upon the difference.

An apple is an abstraction. It's a combination of chemicals bonded together. Those chemicals are composed of atoms which are bonded together, each of which are composed of electrons, protons, and neutrons. Those particles are composed of quarks, and so on.

An apple is a singular thing because it is useful for us to think of an apple as a singular thing in most contexts. It is not a singular thing when it is more useful to look deeper.

Then take quarks. They are a fundamental particle with no known substructure. They are a single unit and they are countable.

Sure, but metaphysically, I’m positing that the whole concept of a unit or particle is a human thing.

Well at least according to the friends I have beer with over the weekends, real world is a subset of math we invented. It is a tool we invented to describe the universe, but I am not sure if everything we derive from the axioms is true. (eg. discontinuous functions make me a little uncomfortable)

Disclaimer: I am very poor at math but eager to learn :)

It is if you round up.

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